You’ll see the following error when trying to connect to a SQL Server Embedded Edition instance, aka Windows Internal Database(MICROSOFT##SSEE instance) using SQL Server Management Studio if you are directly using the server name.
Cannot connect to MachineName\MICROSOFT##SSEE.
A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible. Verify that the instance name is correct and that SQL Server is configured to allow remote connections. (provider: SQL Network Interfaces, error: 26 - Error Locating Server/Instance Specified) (Microsoft SQL Server, Error: -1)
To connect to the Windows Internal Database using Management Studio, specify the server name as \\.\pipe\MSSQL$MICROSOFT##SSEE\sql\query
EDIT: On Windows Server 2012, it seems that we'll need to use \\.\pipe\MICROSOFT##WID\tsql\query. I haven't tried this as I don't have a Windows Server 2012 machine yet, but it is based on the comment below from Stephan.
Hope this helps!